Stability theory of discrete-time dynamical systems involving non-monotonic Lyapunov functions

Abstract

The Lyapunov theorem concerning the global uniform asymptotic stability of an equilibrium x = 0 of a dynamical system involves the existence of scalar-valued, positive definite, radially unbounded, and decrescent functions (of the state and time), called Lyapunov functions, which when
evaluated along the motions of the dynamical system decrease monotonically with increasing time and tend to zero as t →∞. Such functions are called monotonic Lyapunov functions. In more recent results, the monotonicity requirement has been relaxed. These results still mandate that along the motions of the system, the Lyapunov functions approach zero as t →∞; however, it is no longer required
that the decrease of the Lyapunov function along the motions has to be monotonic. Such functions are called non-monotonic Lyapunov functions.
In the present paper we contrast stability results involving monotonic and non-monotonic Lyapunov functions for discrete-time dynamical systems. We show that in general the former are more conservative than the latter. To simplify our presentation, we confine ourselves to discrete-time, finite-
dimensional dynamical systems. In addition to the Lyapunov result for uniform global asymptotic stability, we also address La Salle-Krasovskii type invariance results.